Stability of unconditional convergence almost everywhere

We will investigate the properties of series of functions which are unconditionally convergent almost everywhere on $[0, 1]$. We will establish the following theorem: If the series $\sum_{k=1}^\infty f_k(x)$ converges unconditionally almost everywhere, then there exists a sequence $\{\beta_k\}_1^\infty$, $\beta_k\uparrow\infty$ such that if $\lambda_k\leqslant\beta_k$, $k=1,2,\dots$, the series $\sum_{k=1}^\infty\lambda_k f_k(x)$ converges unconditionally almost everywhere.